Cauchy showed that if the faces of a convex polyhedron are rigid then the
whole polyhedron is rigid. Connelly showed that this is true even if finitely
many extra creases are added. However, cutting the surface of the polyhedron
destroys rigidity and may even allow the polyhedron to be flattened. We
initiate the study of how much the surface of a convex polyhedron must be cut
to allow continuous flattening with rigid faces. We show that a regular
tetrahedron with side lengths 1 can be continuously flattened with rigid faces
after cutting a slit of length .046 and adding a few extra creases.